Results 1 to 3 of 3

Math Help - intersections of subgroups

  1. #1
    Member
    Joined
    Mar 2008
    Posts
    85

    intersections of subgroups

    Let A be a group with normal subgroups M and N.
    Let a \in A.
    Suppose $M \cap \langle a \rangle =1$ and $N \cap \langle a \rangle= \langle a^{s} \rangle$ for some positive integer s.
    Can we find $MN \cap \langle a \rangle$???
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor Drexel28's Avatar
    Joined
    Nov 2009
    From
    Berkeley, California
    Posts
    4,563
    Thanks
    21

    Re: intersections of subgroups

    Quote Originally Posted by deniselim17 View Post
    Let A be a group with normal subgroups M and N.
    Let a \in A.
    Suppose $M \cap \langle a \rangle =1$ and $N \cap \langle a \rangle= \langle a^{s} \rangle$ for some positive integer s.
    Can we find $MN \cap \langle a \rangle$???
    I'll try to help more if this doesn't work, but I have a hunch that this is a question arising from a larger question. If so, can you give it to us--the context will help.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor Swlabr's Avatar
    Joined
    May 2009
    Posts
    1,176

    Re: intersections of subgroups

    Quote Originally Posted by deniselim17 View Post
    Let A be a group with normal subgroups M and N.
    Let a \in A.
    Suppose $M \cap \langle a \rangle =1$ and $N \cap \langle a \rangle= \langle a^{s} \rangle$ for some positive integer s.
    Can we find $MN \cap \langle a \rangle$???
    I am not entirely sure what you are asking...I mean, given finite time of course you can! On the other hand, there isn't a definitive answer. For example, take,

    G=C_4\times C_4\times C_4 ( C_4 denoting the cyclic group of order 4), with your elements being of the form (i, j, k) with 0\leq i, j, k<4. Now, let a=(0, 1, 1). Clearly, if M=\langle (2, 0, 0)\rangle then M\cap \langle a\rangle=1, while if N=\langle(0, 2, 2), (0, 2, 1)\rangle then N\cap\langle a\rangle=\langle (0, 2, 2)\rangle=\langle a^2\rangle. Clearly, MN\cap \langle a\rangle=\langle a^2\rangle. On the other hand, if we let M=\langle(2, 0, 0), (0, 1, 0)\rangle then MN contains (0, 1, 1) but M\cap \langle a\rangle=1 still...so, basically, it depends on M and N and \langle a\rangle, and can be very different depending on different choices...
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Subgroups and Intersection of Normal Subgroups
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: December 1st 2010, 08:12 PM
  2. subgroups and normal subgroups
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: March 19th 2010, 03:30 PM
  3. Intersections
    Posted in the Trigonometry Forum
    Replies: 3
    Last Post: February 19th 2010, 11:34 AM
  4. Subgroups and Normal Subgroups
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: December 9th 2009, 08:36 AM
  5. Subgroups and normal subgroups
    Posted in the Advanced Algebra Forum
    Replies: 8
    Last Post: October 13th 2007, 04:35 PM

Search Tags


/mathhelpforum @mathhelpforum